Romanov's Integral

Calculus Level 5

This integral comes from my personal research and IMHO, it should be a level 5 problem. Why? First, neither Wolfram|Alpha nor Mathematica can return a plausible closed-form for this integral. Not only a closed-form that they failed to give, but also its numerical value to the precision of only 10 digits (using normal procedure). Second, it has been posted at four different sites such as: Mathematics StackExchange, Integrals and Series, AoPS, and Quora, but none exact solution has been given yet. So, if you could answer it, would you care to post the solution? Here is the problem:


\[\begin{equation} \large\mathscr{R}=\int_0^{\Large\frac{\pi}{2}}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx=\frac{\pi}{\alpha}\ln\left(\frac{e^\beta-\gamma}{\delta}\right)-\frac{\pi}{\mu}\left(\frac{e^\nu-\theta}{e^\lambda-\psi}\right)+\omega \end{equation}\]

where \(\large \alpha,\beta,\gamma,\delta,\mu,\nu,\theta,\lambda,\psi,\omega\) are non-negative integers and square-free. Find \(\large \alpha+\beta+\gamma+\delta+\mu+\nu+\theta+\lambda+\psi+\omega\)?

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